where \(d_{\tau \sigma }^{\dagger }\)  creates a hole with energy \(\epsilon _{\tau }\)  in the \(d\)  orbital \(\tau \) , \(n_{\tau \sigma }=d_{\tau \sigma }^{\dagger }d_{\tau \sigma }\) , and \(n_{\tau }=\sum _{\sigma }n_{\tau \sigma }\) .
\(\epsilon _\tau \)  and \(U_\tau \)  are the energy level and the Coulomb
repulsion chosen such that \(\epsilon _\tau = -U_\tau /2\)  to fix the
occupancy in each orbital to one, while the hopping \(V_{\tau }\)  characterizes the tunneling between the
localized and conduction states with symmetry \(\tau \) . The Hund's coupling \(J_H\) 
is responsible for the formation of the \(S=1\)  degree of freedom.
The actual Coulomb interaction contains more terms [1]}, [2]}, [3]},
but they are irrelevant for realistic parameters for FePc.
The two models are
related by the Schrieffer-Wolff transformation, such that \(J_\tau \propto V_\tau ^2/U_\tau \)  (further details are given in Supplementary Note 1).
